Lecture 18:
Second derivatives and concavity.
Analysis of polynomial and rational functions.
Victoria LEBED, [email protected]
MA1S11A: Calculus with Applications for Scientists
November 21, 2017
1 Sign of the second derivative
We have seen that f ′ captures many important properties of f, namely
1) its monotony intervals;
2) its local extrema.
If f is twice di�erentiable, then f ′′ can refine the information given by f ′, as
in the second derivative test. But it also encodes some finer properties of f,
such as the curvature of its graph.
Example. The graphs of the functions f(x) = x2 and f(x) =√x on [0, 1]
share many similarities: both functions are increasing, and take the same
values at endpoints.
x
y y = x2
x
y y =√x
However, the first graph gets steeper as x increases, and the second one gets
less steep. In other words, the graphs are curved di�erently.
1 Sign of the second derivative
To talk about the curvature rigorously, we need the following terms.
Definition. A function f di�erentiable on (a, b) is called
X concave up (or convex) if f ′ is increasing on (a, b);
X concave down (or concave) if f ′ is decreasing on (a, b).
Informally, a function is
X concave up if it holds water;
X concave down if it spills water.
x
yy = x2
x
yy = 1− x2
concave up concave down
1 Sign of the second derivative
There are two alternative definitions of a concave up/down function:
2) a function lying above/below any of its tangents;
3) a function lying below/above any of its chords.
concave up concave down
These definitions compare the function analysed with linear functions.
Definition 3) works for not necessarily di�erentiable functions.
1 Sign of the second derivative
Theorem 7. Let f be a twice di�erentiable function on (a, b).
(a) If f ′′(x) > 0 on (a, b), then f is concave up.
(b) If f ′′(x) < 0 on (a, b), then f is concave down.
This is true because of the connection between the monotony and the sign
of the derivative (Theorem 4), applied to f ′.
Example 1.
x
yy = x2
x
yy = 1− x2
f ′′(x) = 2 > 0 f ′′(x) = −2 < 0
concave up concave down
Example 2. For the function f(x) = x2 − 6x+ 5, we have f ′(x) = 2x− 6 and
f ′′(x) = 2, so this function is concave up on R = (−∞,+∞).
2 Zeroes of the second derivative
A function seldom has the same concavity type on its whole domain.
Definition. An inflection point of a function f is a point where it changes
the direction of concavity.
In other words, an inflection point marks the places on the curve y = f(x)
where the rate of change of y with respect to x (that is, f ′) changes from
increasing to decreasing, or vice versa.
Theorem 8. If f is a twice di�erentiable at c, and c is an inflection point for
f, then f ′′(c) = 0.
For example, if f(x) is the height of the water
level in a vase when it contains x units of
water, the inflection points of the graph of
f(x) correspond to heights at which the flask
is the most narrow, or the most wide.
2 Zeroes of the second derivative
Definition. An inflection point of a function f is a point where f changes
the direction of concavity.
Theorem 8: c is an inflection point =⇒ f ′′(c) = 0.
Example 3. For the function f(x) = x3, we have f ′(x) = 3x2 and f ′′(x) = 6x,
so this function is concave up on (0,+∞), and is concave down on (−∞, 0).
x
yy = x3
b
f ′′(x) < 0 f ′′(x) = 0 f ′′(x) > 0
concave down inflection point concave up
2 Zeroes of the second derivative
Example 4. Let us analyse the inflection points of the function
f(x) = x2− sin x on [0, 2π] that we considered earlier. We have
f ′(x) =1
2− cos x, f ′′(x) = sinx.
0 (0, π) π (π, 2π) 2π
f ′′(x) = sin x 0 > 0 0 < 0 0
concavity up inflection point down
2 Zeroes of the second derivative
Example 4 (continued). The information gathered about f(x) = x2− sin x on
[0, 2π] (monotony, local and global extrema, behaviour at endpoints,
concavity up and down, inflection points) allows us to plot an approximate
graph of f:
x
y
π/3 π 5π/3
3 Analysis of polynomial functions
From the viewpoint of calculus, polynomials are the simplest functions:
X their natural domain is R;
X they are continuous and di�erentiable everywhere;
X they increase or decrease without bound when x → ±∞ (depending
on the leading term);
X a polynomial of degree n has at most
• n roots (i.e., zeroes, or x-intercepts);
• n − 1 local extrema;
• n − 2 inflection points;
X a polynomial of odd degree n has no global extrema on R;
X a polynomial of even degree n has either global maximum but no
global minimum on R, or the other way round.
x
y
x
y
3 Analysis of polynomial functions
But polynomials are not entirely boring. Possibly the most interesting part
about graphing polynomials is the geometric interpretation of the
multiplicities of their roots (which is a priori an algebraic notion):
Definition. Given a polynomial P, a real number r is called a root of P of
multiplicity m if (x− r)m divides P(x) but (x − r)m+1 does not. In the
case m = 1, r is called a simple root.
When we say that a polynomial of degree n has at most n real roots, we
count each root with multiplicities.
Theorem 8. Suppose that r is a root of P(x) of multiplicitym. Then
parity of m position of the x-axis inflection point
w.r.t. the graph of P at x = r at x = r?
even tangent but not crossing no
odd > 1 tangent and crossing yes
odd = 1 not tangent and crossing yes or no
4 Analysis of rational functions
Analysis of rational functions is more involved, since there are variations in
what the domain is, what the asymptotes are etc. The algorithm on the
following slides gives a way to identify the most important features of the
graph of a rational function, and to sketch it.
We shall deal only with rational functions
in the reduced form f(x) =P(x)Q(x)
, i.e.,
the polynomials P andQ will have no
common factors. If they do, we can
cancel all common factors, remember
that f is actually not defined where those
factors vanish, and remove the respective
points from the graph, as we did before.
x
y
bc
4 Analysis of rational functions
Definition. Given a rational function in the reduced form
f(x) = P(x)/Q(x), a real number r is called
X a root of f of multiplicity m if it is a root of P of multiplicitym.
X a pole of f of multiplicity m if it is a root of Q of multiplicitym.
The behaviour of a rational function close to its roots is the same as for
polynomials.
At a pole r of multiplicitym, f has a vertical asymptote, and the sign of f
changes at r (from +∞ to −∞ or the other way round) i�m is odd.
4 Analysis of rational functions
Algorithm for graphing f(x) = P(x)/Q(x) (for P andQ without
common factors).
X Determine if the graph has symmetries about the y-axis / the origin,
i.e., whether f is even/odd.
X Find where and how the graph meets the x-axis, i.e., compute the roots
of f and their multiplicities.
X Find where the graph meets the y-axis, i.e., compute f(0).
X Determine all vertical asymptotes and check if there is a sign change
across them, i.e., compute the poles of f and their multiplicities.
X Compute the limits of f(x) at ±∞. This yields the horizontal
asymptote of the graph, if any.
X Determine the sign of f on each interval between the x-intercepts and
the vertical asymptotes.
X Determine where f is increasing/decreasing, concave up/down. Find all
critical points, local and global extrema, inflection points. For this,
analyse the signs of f ′ and f ′′, if they exist.
X Sketch the graph.
5 Analysis of rational functions: example
Let us analyse the rational function f(x) = x2−1x3 .
X Reduced form:x2 − 1
x3,
since x2 − 1 = (x− 1)(x + 1) and x3 have no common factors.
X Symmetries: f(−x) =x2 − 1
−x3= −
x2 − 1
x3= −f(x) =⇒ f is odd
=⇒ its graph is symmetric about the origin.
X x-intercepts: f has two roots of multiplicity 1: ±1 =⇒ its graph
intersects the x-axis at the points ±1, where the graph changes sign.
X y-intercepts: none, as the point 0 is not in the natural domain of f.
X Vertical asymptotes: f has one pole, 0, of multiplicity 3: =⇒ at 0 there
is a vertical asymptote, and f changes sign.
5 Analysis of rational functions: example
Let us analyse the rational function f(x) = x2−1x3 .
X Horizontal asymptotes: We have
limx→+∞
x2 − 1
x3= lim
x→+∞
(
1
x−
1
x3
)
= 0,
limx→−∞
x2 − 1
x3= 0 by symmetry,
so y = 0 is the only horizontal asymptote.
X Sign analysis: as seen above, f changes sign at the points −1, 0, 1; since
f(2) = 18 > 0, f(x) > 0 on (1,+∞), which then yields the sign of f
everywhere:
interval (−∞,−1) (−1, 0) (0, 1) (1,+∞)
sign of f − + − +
5 Analysis of rational functions: example
Let us analyse the rational function f(x) = x2−1x3 .
X Monotony and extrema: the first derivative
f ′(x) = (x−1−x−3) ′ = −x−2+3x−4 =3− x2
x4=
(√3− x)(x +
√3)
x4
vanishes at x = ±√3; signs of the corresponding factors result in the
following signs for f ′:
interval (−∞,−√3) (−
√3, 0) (0,
√3) (
√3,+∞)
signs of (√3− x) (+)(−) (+)(+) (+)(+) (−)(+)
and (x +√3)
sign of f ′ − + + −
So, f is decreasing on (−∞,−√3] and on [
√3,+∞) and is increasing
on [−√3, 0) as well as on (0,
√3]. Therefore, it has a local minimum at
x = −√3 and a local maximum at x =
√3.
There is no global minimum or maximum, since limx→0+
= −∞ and
limx→0−
= +∞.
5 Analysis of rational functions: example
Let us analyse the rational function f(x) = x2−1x3 .
X Concavity: the second derivative
f ′′(x) = (f ′(x)) ′ = (−x−2 + 3x−4) ′ = 2x−3 − 12x−5
=2(x2 − 6)
x5=
2(x −√6)(x +
√6)
x5
vanishes for x = ±√6;
signs of the corresponding factors result in the following signs for f ′′:
interval (−∞,−√6) (−
√6, 0) (0,
√6) (
√6,+∞)
signs of factors
x5, (x−√6), (−)(−)(−) (−)(−)(+) (+)(−)(+) (+)(+)(+)
and (x +√6)
sign of f ′′ − + − +
This suggests that f is concave down on (−∞,−√6] and on (0,
√6]
and is concave up on [−√6, 0) and on [
√6,+∞).
Therefore, x = ±√6 are inflection points.
5 Analysis of rational functions: example
Summarising all the available information, we plot the graph of
f(x) = x2−1x3 as follows:
x
y
b
b
b
b
b
b
b b
6 Curvilinear asymptotes
A curvilinear asymptote is a curve that approaches a given curve
arbitrarily closely.
They are typically used for describing the behaviour of the graph of a
function f(x)
X at its poles (this is what vertical asymptotes do);
X at ±∞ (this is what horizontal/oblique asymptotes do).
We will now look at a particular type of asymptotes. If for a function f(x) of
interest you can find another (preferably simpler) function g(x) such that
f(x) − g(x) →x→+∞
0 or f(x) − g(x) →x→−∞
0,
then the graph of g is an asymptote of the graph of f. This means that you
can understand the behaviour of f for very big/small x by looking at the
simpler function g.
We have seen two particular cases of this situation:
X g(x) = c is a constant function (and defines a horizontal asymptote);
X g(x) = bx+ c is a linear function (and defines an oblique asymptote).
6 Curvilinear asymptotes
To describe a rational function f(x) =P(x)Q(x)
at ±∞, it is helpful to
determine a polynomial function which yields its curvilinear asymptote.
For this, you need the polynomial long division. It allows you to write
P(x) = Q(x)S(x) + R(x),
where S and R are polynomials, and the degree of R is less than the degree
of Q. As a result,R(x)Q(x)
→x→±∞
0 (recall our earlier discussion on the limits of
rational functions). So,P(x)
Q(x)− S(x) =
P(x) −Q(x)S(x)
Q(x)=
R(x)
Q(x)→
x→±∞
0.
This means that S(x) is the “principal part” ofP(x)Q(x) , and the graph of S(x)
is a curvilinear asymptote for the graph ofP(x)Q(x) .
So, a rational function always has a curvilinear asymptote, and this
asymptote is the same at +∞ and at −∞.
6 Curvilinear asymptotes
Example 1. f(x) = x4−2x3+3x+4x
=x(x3−2x2+3)+4
x= x3 − 2x2 + 3+ 4
x.
So, the graph of x3 − 2x2 + 3 is an asymptote to the graph of x4−2x3+3x+4x
.
x
y
6 Curvilinear asymptotes
Example 2. f(x) = x4−2x2−1
=(x2−1)(x2+1)−1
x2−1= x2 + 1− 1
x2−1.
So, the graph of x2 + 1 is a parabolic asymptote to the graph of x4−2x2−1
.
